This paper develops the theory of the design and performance of optimal finite-memory systems for the two-hypothesis testing problem. Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables drawn according to a probability measure $\mathscr{P}$. Consider the standard two-hypothesis testing problem with probability of error loss criterion in which $\mathscr{P} = \mathscr{P}_0$ with probability $\pi_0$; and $\mathscr{P} = \mathscr{P}_1$ with probability $\pi_1$. Let the data be summarized after each new observation by an $m$-valued statistic $T\in\{ 1, 2, \cdots, m\}$ which is updated according to the rule $T_n = f(T_{n-1}, X_n),$ where $f$ is a (perhaps randomized) time-invariant function. Let $d:\{ 1, 2,\cdots, m\} \rightarrow\{ H_0, H_1\}$ be a fixed decision function taking action $d(T_n)$ at time $n$, and let $P_e(f,d)$ be the long-run probability of error of the algorithm $(f, d)$ as the number of trials $n\rightarrow\infty$. Define $P^\ast = \inf_{(f,d)}P_e(f, d)$. Let the a.e. maximum and minimum likelihood ratios be defined by $\bar{l} = \sup(\mathscr{P}_0(A)/\mathscr{P}_1(A))$ and $\underline{l} = \inf(\mathscr{P}_0(A)/\mathscr{P}_1(A))$ where the supremum and infimum are taken over all measurable sets $A$ for which $\mathscr{P}_0(A) + \mathscr{P}_1(A) > 0$. Define $\gamma = \bar{l}/\underline{l}$. It will be shown that $P^\ast = \lbrack 2(\pi_0\pi_1\gamma^{m-1})^{\frac{1}{2}} - 1\rbrack/(\gamma^{m-1} - 1)$, under the nondegeneracy condition $\gamma^{m-1} \geqq \max\{\pi_0/\pi_1, \pi_1/\pi_0\}$; and a simple family of $\varepsilon$-optimal $(f, d)$'s will be exhibited. In general, an optimal $(f, d)$ does not exist; and $\varepsilon$-optimal algorithms involve randomization in $f$.

Consider a statistical procedure (Method $A$) which is based on $n$ observations, and a less effective procedure (Method $B$) which requires a larger number $k_n$ of observations to give equally good performance. Comparison of the two methods involves the comparison of $k_n$ with $n$, and this can be carried out in various ways. Perhaps the most natural quantity to examine is the difference $k_n - n$, the number of additional observations required by the less effective method. Such difference comparisons have been performed from time to time. (See, for example, Fisher (1925), Walsh (1949) and Pearson (1950).) Historically, however, comparisons have been based mainly on the ratio $k_n/n$. Thus, Fisher (1920), in comparing the mean absolute deviation with the mean squared deviation as estimates of a normal scale, found this ratio to be 1/1.14. Similarly in 1925 he found a large-sample ratio of $2/\pi$ for median compared with mean for estimating normal location, and the same value was found by Cochran (1937) for the sign test relative to the $t$-test in the normal case. The reason for using the ratio rather than the difference in these cases is of course that the ratio is stable in large samples, so that a single limit value, say $e = \lim_{n\rightarrow\infty} n/k_n$, known as the asymptotic relative efficiency or ARE of $B$ with respect to $A$, conveys a great deal of useful information in a compact form. When $e < 1$, the difference $k_n - n$, is not a useful measure in large samples because it tends to infinity with $n$. The situation is however different, and in a sense even the reverse, in the many important statistical problems in which $e = 1$. It is then possible also for the difference to be stable, and the main purpose of this paper is to point out a number of problems in which this is the case. For the additional number $k_n - n$ of observations needed by Method $B$ we suggest the term deficiency. If it exists, the limit value $d = \lim_{n\rightarrow\infty}(k_n - n)$, will be called the asymptotic deficiency. The number $d$ summarizes the comparison much more revealingly in these cases than does the fact that $e$ equals 1. (If it is not known a priori, the latter does not even tell us which of the two procedures is better.) Of course, $d$ is less easy to compute than $e$ since it in effect requires computing an additional term in the asymptotic expansions.

In this paper, an extension of the investigation of Johnson (1967b) is made by giving a larger class of posterior distributions which possess asymptotic expansions having a normal distribution as a leading term. Asymptotic expansions for the related normalizing transformation and percentiles are also presented. Before asymptotic expansions were treated rigorously, LaPlace (1847) gave an expansion for certain posterior distributions. The method used in this paper is a variation of his technique. Bernstein (1934), page 406, and von Mises (1964), chapter VIII, Section C, also treat special cases of these expansions. The conditions imposed are sufficient to make the maximum likelihood estimate strongly consistent and asymptotically normal. They also include higher order derivative assumptions on the log of the likelihood. As shown by Schwartz (1966), the posterior distribution may behave well even when the maximum likelihood estimate does not. However, we have not attempted to find the weakest assumptions under which the posterior distribution has an expansion. For general conditions under which the posterior distribution converges in variation to a normal distribution with probability one see LeCam (1953) and (1958) for the independent case and Kallianpur and Borwanker (1968) for Markov processes. In Section 2, we show that with probability one, the centered and scaled posterior distribution possesses an asymptotic expansion in powers of $n^{-\frac{1}{2}}$ having the standard normal as a leading term. The number of terms in the expansion obtained is two less than the number of continuous derivatives of the log likelihood. All terms beyond the first consist of a polynomial multiplied by the standard normal density. The coefficients of the polynomial depend on the prior density $\rho$ and the likelihood. The moments of the posterior distribution are shown to possess an expansion in Section 3. The following two sections present the normalizing transformation and percentile expansions. These last three expansions also apply for the case considered by Johnson (1967b) as does the information on the form of the terms in the expansion of the posterior distribution. To simplify the already heavy notation, these results are first proved for independent identically distributed random variables. The extension of all these results to the case of certain stationary ergodic Markov processes is immediate; Section 6 presents the necessary modifications. Throughout this paper, $\Phi$ and $\varphi$ will denote the standard normal cdf and pdf respectively. Also, $\mathbf{n}$ will be assumed to range over the positive integers; thus in some cases, the order of the error term in the expansion may be kept for smaller $n$ if the bounding constant is modified.

Conditions are given under which a sequence of posterior distributions converges weakly to a degenerate distribution. It is not assumed that the model being used actually governs the data. Exponential models are studied in particular and are shown to be well behaved under mild assumptions. Large-deviation results for $U$-statistics and posterior odds are also given.

Deming and Stephan (1940) first proposed the use of an iterative proportional fitting procedure to estimate cell probabilities in a contingency table subject to certain marginal constraints. In this paper we first relate this procedure to a variety of sources and a variety of statistical problems. We then describe the procedure geometrically for two-way contingency tables using the concepts presented in Fienberg (1968). This geometrical description leads to a rather simple proof of the convergence of the iterative procedure. We conclude the paper with a discussion of extensions to multi-dimensional tables and to tables with some zero entries.

Let $(\mathscr{X}, \mathscr{A})$ be a measurable space and let $\Theta$ be an open subset of the $k$-dimensional Euclidean space $\mathscr{E}_k$. For each $\theta\in\Theta$, let $P_\theta$ be a probability measure on $\mathscr{A}$. Let $\{ X_n, n \geqq 0\}$ be a discrete parameter Markov process defined on $(\mathscr{X}, \mathscr{A}, P_\theta), X_n$ taking values in the Borel real line $(R, \mathscr{B})$. Finally, let $\mathscr{A}_n$ be the $\sigma$-field induced by the first $n + 1$ random variables $X_0, X_1,\cdots, X_n$ from the process and let $P_{n,\theta}$ be the restriction of $P_\theta$ to the $\sigma$-field $\mathscr{A}_n$. Under suitable conditions on the process, the following results are derived. Let $\theta_0$ be an arbitrary but fixed point in $\Theta$ and let $\Delta_n(\theta_0)$ be a $k$-dimensional vector defined in terms of the random variables $X_0, X_1,\cdots, X_n; \Delta_n^\ast(\theta_0)$ stands for a certain truncated version of $\Delta_n(\theta_0)$. By means of $\Delta_n^\ast(\theta_0)$ and $h\in\mathscr{E}_k$, one defines a probability measure $R_{n,h}, n \geqq 0$. The first main result is that the sequences $\{P_{n,\theta}\}$ and $\{R_{n,h}\}$ of probability measures with $h = n^{\frac{1}{2}}(\theta - \theta_0), \theta\in\Theta$, are differentially equivalent at the point $\theta_0$. (See Definition 5.1.) This is shown in Corollary 5.1. It is also proved in Corollary 5.2 that the sequence $\{\Delta_n^\ast(\theta_0)\}$ is differentially sufficient at $\theta_0$ (see Definition 5.2) for the family $\{P_{n,\theta}; \theta\in\Theta\}$ of probability measures. Next, let $\{h_n\}$ be a bounded sequence of $h$'s in $\mathscr{E}_k$ and set $\theta_n = \theta_0 + h_nn^{-\frac{1}{2}}$. Then for hypotheses testing problems, Theorem 6.1 allows one to restrict oneself to the class of tests depending on $\Delta_n(\theta_0)$ alone, at least as far as the asymptotic power of the test under alternatives of the form $P_{n,\theta_n}$ is concerned. In Section 7, these results are applied to the case of testing hypotheses about a real-valued parameter. More specifically, asymptotically most powerful tests for testing the hypothesis $\theta = \theta_0$ against one-sided alternatives are constructed. This is covered in Theorem 7.1.1. Also an asymptotically most powerful unbiased test for testing the same hypothesis as above against two-sided alternatives is constructed in Theorem 7.1.2. The first of these problems was also dealt with in Johnson and Roussas [2] but the approach is different here. The second problem is solved in Wald [8] for the independent identically distributed case. However, both the assumptions and approach are different here in addition to the Markovian character of the random variables involved. Section 6 treats the general situation where $\Theta$ is an open subset of $\mathscr{E}_k$. Theorem 6.1 together with Theorem 6.3 provide a way for studying the corresponding hypothesis testing problem in the $k$-dimensional parameter case. Finally, at the end of the last section, an outline is presented of forthcoming results for that case. These results extend, under substantially weaker conditions, the work of Wald [8], [9] to Markov processes. The method of proof relies heavily on the development in LeCam [3]. Unless otherwise stated, limits will be taken as the sequence $\{ n\}$, or subsequences thereof, tends to $\infty$. Integrals without limits will extend over the entire appropriate space. For $h\in\mathscr{E}_k, h'$ stands for its transpose. All bounding constants will be finite numbers.

The response variables of a first order autoregressive stochastic process with autocorrelation $\rho$ can be constructed as a location-scale transform of a set of error variables whose distribution depends on the autocorrelation parameter, and the model can be treated as a composite response model with an error quantity $\rho$ (Fraser 1968, page 192). For known value of $\rho$ the model is a conditional structural model (Fraser 1968, page 188). Inference concerning $\rho$ is naturally based on the marginal likelihood function of $\rho$ obtained from the marginal probability distribution of the orbit of the response. The results are specialized to cover the normal error distribution. For the same model with error variables having a periodic structure a suitable transformation reduces the error variable into uncorrelated variables and the resultant transformed model can be treated as a location-scale structural model. The orbit of the transformed response depends on $\rho$ and so inference concerning $\rho$ can be made from the marginal likelihood of $\rho$ obtained from the marginal probability distribution of the orbit. It has been found that without any approximation being used the marginal likelihood function of $\rho$ thus obtained depends on the first order circular serial correlation coefficient in general, and for normal distribution in particular. For both the cases the general approximate distribution of the serial correlation coefficient has been derived by likelihood modulation. Using Anderson's (1942) result the general exact distribution of the circular serial correlation coefficient has also been obtained.

Let $\mathbf{X}_1,\mathbf{X}_2,\cdots$ be a stochastic sequence and $\mathscr{P}$ and $\mathscr{L}$, two composite parametric hypotheses (models) under which the $\mathbf{X}_i$ are i.i.d. We consider SPRTs of $\mathscr{P}$ vs $\mathscr{L}$ that depend on a sequence of exchangeable densities. Included are SPRTs obtained by the method of weight-functions (Bayesian procedures) and many SPRTs obtained by invariance reduction. Conditions are established under which the stopping time of such a procedure is almost surely finite and has a nontrivial mgf. The ideas are illustrated using the sequential $t$-test.

Many distributions in multivariate analysis can be expressed in a form involving hypergeometric functions $_pF_q$ of matrix argument e.g. the noncentral Wishart $(_0F_1)$ and the noncentral multivariate $F(_1F_1)$. For an exposition of distributions in this form see James [9]. The hypergeometric function $_pF_q$ has been defined by Constantine [1] as the power series representation \begin{equation*}\tag{1.1} _pF_q(a_1,\cdots, a_p; b_1,\cdots, b_q; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a_1)_\kappa\cdots(a_p)_\kappa}{(b_1)_\kappa\cdots (b_q)_\kappa} \frac{C_\kappa (R)}{k!}\end{equation*} where $a_1,\cdots, a_p, b_1,\cdots, b_q$ are real or complex constants, $(a)_\kappa = \mathbf{prod}^m_{i=1}(a - \frac{1}{2}(i - 1))_{k_i},\quad (a)_n = a(a + 1)\cdots (a + n - 1)$ and $C_\kappa(R)$ is the zonal polynomial of the $m \times m$ symmetric matrix $R$ corresponding to the partition $\kappa = (k_1, k_2,\cdots, k_m), k_1 \geqq k_2 \geqq \cdots \geqq k_m$, of the integer $k$ into not more than $m$ parts. The functions defined by (1.1) are identical with the hypergeometric functions defined by Herz [5] by means of Laplace and inverse Laplace transforms. For a detailed discussion of hypergeometric functions and zonal polynomials, the reader is referred to the papers [1] of Constantine and [7], [8], [9] of James. From a practical point of view, however, the series (1.1) may not be of great value. Although computer programs have been developed for calculating zonal polynomials up to quite high order, the series (1.1) may converge very slowly. It appears that some asymptotic expansions for such functions must be obtained. It is well known that asymptotic expansions for a function can in many cases be derived using a differential equation satisfied by the function (see e.g. Erdelyi [4]), and so, with this in mind, a study of differential equations satisfied by certain hypergeometric functions certainly seems justified. In this paper a conjecture due to A. G. Constantine is verified i.e. it is shown that the function \begin{equation*}\tag{1.2} _2F_1(a, b; c; R) = \sum^\infty_{k=0} \sum_\kappa \frac{(a)_\kappa(b)_\kappa}{(c)_\kappa} \frac{C_\kappa(R)}{K!}\end{equation*} satisfies the system of partial differential equations \begin{align*} \tag{1.3} R_i(1 &- R_i)\partial^2F/\partial R_i^2 + \{ c - \frac{1}{2}(m - 1) - (a + b + 1 - \frac{1}{2}(m - 1))R_i \\ &+\frac{1}{2} \sum^m_{j=1,j\neq i}\lbrack R_i(1 - R_i)/(R_i - R_j) \rbrack\}\partial F/\partial R_i \\ &-\frac{1}{2} \sum^m_{j=1,j\neq i} \lbrack R_j(1 - R_j)/(R_i - R_j) \rbrack\partial F/\partial R_j = abF \quad (i = 1,2, \cdots, m)\end{align*} where $R_1, R_2,\cdots, R_m$ are the latent roots of the complex symmetric $m \times m$ matrix $R$. When $m = 1$, the system (1.3) clearly reduces to the classical hypergeometric equation. It appears difficult to establish this conjecture directly, and the method used has necessitated a section devoted to a summary of the argument involved (Section 3). The main result in the paper is summarized in Theorem 3.1 of this section. Section 4 contains proofs referred to in Section 3. Using the fact that $C_\kappa (R)$ satisfies the partial differential equation (James [10])\begin{equation*}\tag{1.4} \sum^m_{i=1} R_i^2\partial^2y/\partial R_i^2 + \sum^m_{j=1,j\neq i}\lbrack R_i^2/(R_i - R_j) \rbrack\partial y/\partial R_i = \sum^m_{i=1}k_i(k_i + m - i - 1)y,\end{equation*} James and Constantine [11] have obtained the effects of certain differential operators on $C_\kappa(R)$. These results are given in Section 2 and are used in many proofs in Section 4. Section 5 is probably of most interest statistically, for here systems of partial differential equations similar to (1.3) are given for $_1F_1(a; c; R)$ and $_0F_1(c; R)$. These two functions occur often in multivariate distributions. The differential equations for $_1F_1(a; c; R)$ have been used by Constantine [3] to obtain an asymptotic expansion for the noncentral likelihood ratio criterion, and by the author [12] to obtain asymptotic distributions of Hotelling's generalized $T_0^2$ statistic, Pillai's $V^{(m)}$ criterion, and for the largest latent root of the covariance matrix. The system for $_0F_1(c; R)$ is a generalization of that given by James [6] for $_0F_1(m/2; R)$.

In this paper it is shown how the systems of partial differential equations developed by the author [15] for the hypergeometric functions of matrix argument may be used to obtain asymptotic expansions for some distributions occurring in multivariate analysis. In particular expansions are derived for the distributions of Hotelling's generalized $T_0^2$-statistic, Pillai's $V^{(s)}$ criterion, and the largest latent root of the sample covariance matrix.

The inference and decision models considered here are those described by Neyman in [5] pages 16 and 17. For a random variable $X$ with possible probability distributions indexed by a parameter $\theta$, Neyman distinguishes two inference approaches to the problem of estimating $\theta$: (i) having observed $X = x$, find the most probable values $z(x)$ of $\theta$ (This requires a priori probabilities, which often must be chosen rather arbitrarily, and whose existence may be questioned.); and (ii) having observed $X = x$, find the values $z(x)$ that are most reasonable or in which we have the greatest confidence (This requires the rather arbitrary choice of a real-valued function $L$, with $L(x, \theta)$ measuring the degree of confidence we have in the parameter value $\theta$ given that $X = x$ has been observed.). Of these two approaches, (i) is a special case of (ii) since in particular $L(x, \theta)$ can be taken to be the a posteriori probability of $\theta$ given $X = x$, for a specified a priori distribution of the parameter. Neyman remarks that in his opinion "the inferential theory solves no problem" and proceeds to describe a real world situation for which the decision model is a very good one. For a random variable $X$ with possible probability distributions indexed by a parameter $\theta$ the decision approach to estimating $\theta$ is to associate, with the use of each possible estimator $z(X)$, a random loss $W(z, X, \theta)$ whose possible distributions are indexed by $\theta$; and to determine $z$ so that this loss will, in some specified sense, be as small as possible on the whole over all $\theta$ values. The purpose of the present paper is to examine Neyman's inference model in detail, to describe real situations for which it appears to be a good model, and to compare these with situations for which the decision model is more appropriate. Mathematical models are described, but there is almost no mathematics in the sense of deriving details for the models: concern is mostly with the applied mathematics question of what mathematical model to use for a real situation. Section 2 is a detailed description of Neyman's general inference model. Here $L(x, \theta)$ is interpreted as a measure of agreement between $P_\theta$ probabilities and observed proportions; terms such as "most probable," "most reasonable," "greatest confidence" are avoided as having connotations that are difficult to support. The general inference estimator is just Wolfowitz's minimum distance estimator [12]. Wolfowitz evaluates such procedures purely from the decision viewpoint. Section 3 is a detailed description of the general decision model. This is given in a form closely paralleling the description of Section 2, in order that the two models can be compared easily. In Section 4, the inference and decision models are compared from several viewpoints. The inference problem seeks an estimator $z$ such that the $P_{z(x)}$ probabilities will be (for all $x$) close to the proportions observed in $x$; the decision problem of estimating $\theta$ seeks an estimator $z$ such that $z(X)$ will be (for all $\theta$) close to $\theta$. The decision problem requires the idea of distance or error, measured by loss, in the parameter space; this idea is completely or partially absent in the inference model, where $\theta$ merely indexes the possible probability models. It is this presence or absence of a loss function that distinguishes between the decision and inference models as defined here: in the decision problem the use of an estimator $z$ results in definite losses and we are to determine a $z$ for which they are small; in the inference model the idea of definite losses does not appear. The decision model is a much more specific model for a much more specific real problem. In both problems we want to choose a probability model for the real situation: in the decision problem we know what the model is to be used for; in the inference problem we do not. It can be reasonably argued, following Neyman [5] page 17, that the inference model is sometimes used in the mistaken belief that making $P_{z(x)}$ close to the observed proportions will make $P_{z(X)}$ close to $P_\theta$, and that if so it should be replaced by the decision model with loss $W\lbrack P_{z(X)}, P_\theta\rbrack$. In the inference problem as described here, we do not have this definite aim of making $P_{z(X)}$ close to $P_\theta$: we are uncertain as to whether we want $P_{z(X)}$ close to $P_\theta$ or want $z(X)$ close to $\theta$ or want $\lbrack z(X)\rbrack^2$ close to $\theta^2$ or want something else; vaguely, we have all decision-type aims, but we have no definite one. Both the inference and decision models require the making of somewhat analogous and rather arbitrary choices at two levels. (Here, and throughout this paper "arbitrary" is used in its primary dictionary meaning of "depending on will or discretion; discretionary; can be freely chosen," with none of the secondary dictionary meanings of "unreasoned, despotic." Possible synonyms such as "subjective," "individualistic," "personalistic" are avoided because of their technical meanings.) Neyman's view that the user should be fully entitled to any choices he cares to make and that no attempt should be made to impose particular choices on all, is followed here. In Section 5 the most commonly used inference methods (Likelihood, Least Squares and Moments, Chi-square, Kolmogorov-Smirnov) are examined as special cases of the general inference method.

Given a sequence $\{X_n; n \geqq 0\}$ of random variables, let $\{P^\theta; \theta \in \pi\}$ be a parameterized class of probability measures, with respect to each of which the sequence is a Markov chain. Under conditions which make it appropriate to define likelihood ratios, the parameter set $\pi$ can be identified with a subset of the Martin boundary for the space-time chain $\{(n, X_n); n \geqq 0\}$, so that each parameter $\theta$ can also be considered as a point of this Martin boundary. Then for each $\theta$, the spacetime sample paths converge in the Martin boundary topology to $\theta$, almost surely with respect to the probability measure $P^\theta$. Moreover, the likelihood ratio corresponding to the parameter $\theta$ is the same as the minimal regular function corresponding to the parameter $\theta$, and the probability measure $P^\theta$ is the relativised probability measure corresponding to the point $\theta$.

A random variable $x$ is said to have the inverse Gaussian [3], or Wald's ([4] pages 192-193) distribution function when its density function is given by \begin{equation*}\tag{1} f(x; m, \lambda) = (\lambda/2\pi x^3)^{\frac{1}{2}} \exp\{-\lambda(x - m)^2/2m^2x\},\quad 0 < x < \infty,\end{equation*} where the constants $\lambda$ and $m$ are assumed to be positive. If $x_1, x_2,\cdots, x_N$ are $N$ independent observations from (1), then it is proved by Tweedie [3] that the densities of the variates $y = \sum^N_{j=1} x_j$ and $z = \sum^N_{j=1} x_j^{-1} - N^2y^{-1}$ are independent; the distribution of $y$ is $f(y; Nm, N^2\lambda)$ and that of $\lambda z$ is $\chi^2$ with $N - 1$ degrees of freedom (df). By proving the converse of Tweedie's result, namely, if $x_1,\cdots, x_N$ are independently and identically distributed random variables and if $y$ and $z$ are independently distributed, then $x_1, x_2,\cdots, x_N$ each have inverse Gaussian distribution, Khatri [2] gave a characterization of the inverse Gaussian distribution. Now Wald ([4] pages 192-193) has proved that if $x$ has the distribution (1), then $t^2 = (x - m)^2/m^2x$ has a $\chi^2$ distribution with one df, or $t$ has a normal distribution with mean zero and variance $\lambda^{-1}$. It thus appears that certain properties of the inverse Gaussian distribution may be studied via the known properties of the normal distribution. Khatri's result on the characterization of the inverse Gaussian distribution is one such result. The following known result for the normal distribution, namely, if $t_1, t_2,\cdots, t_N$ are independent and identically distributed random variables and if a linear form $w$ in $t$ variates and a quadratic form $z$ of rank $N - 1$ in $t$ variates, where $\sum^N_1 t^2_i = w^2 + z$, are independently distributed, then the $t$ variates are normally distributed. Now this result when translated for the inverse Gaussian distribution is the result stated by Khatri, who proves the result by an involved method of characteristic functions. Perhaps, the simpler proof of Khatri's result established in the present paper might be of pedagogical interest.

Unbiased estimators are derived for a sample from the geometric density with unknown $p$ and unknown location parameter. Mean square errors are compared with the maximum likelihood estimator and unbiased tests of hypotheses are given.

The listing of partially balanced linked block designs by Roy and Laha (1957) contains no Latin square designs. The listing of designs with Latin square association schemes by Clatworthy (1956), which includes those given by Bose, Clatworthy and Shrikhande (1954) and by Bose and Shimamoto (1952), and the later listings by Chang and Liu (1964) and by Clatworthy (1967) contain no linked block designs. The question then arises whether any linked block designs exist having the Latin square association scheme. In this note a partial answer to the question is given. It is shown that there do not exist any linked block designs which are partially balanced with two associate classes and have the $L_i$ association scheme for $i = 2,3$ or 4.

In a recent paper [3], White and Hultquist extended the use of finite fields for the construction of confounding plans to include "asymmetrical" or "mixed" factorials. The technique, in their own words, was to define addition and multiplication of elements from distinct finite fields by mapping these elements into a finite commutative ring containing subrings isomorphic to each of the fields in question. The standard techniques were then applied to the asymmetrical case illustrating the procedure with a numerical example for $3^2 \times 5$ factorial experiment. Later, Raktoe [1] also provided an equivalent theoretical basis for the results obtained by White and Hultquist [3], worked out a generalization of the technique and illustrated his procedure with an example of $2^2 \times 3 \times 5$ factorial experiment. It appears, however, that to provide a basis of the required calculus covering confounding plans of "mixed factorials" of the types discussed by them, it may not be necessary to invoke the properties of finite fields and to combine them. Instead, properties of finite multiplicative groups may be sufficient to construct such confounding plans. The aim of the present note is to indicate that this alternative approach, when it exists, is structurally identical with the procedure as outlined in [3], and that this methodology is simple, taking as it does only the properties of multiplicative groups. The procedure has been illustrated with reference to the same example of $3^2 \times 5$ factorial design as discussed in full in [3]. The correspondence relationships between the levels of the factors and the elements of the group may be so worked out that the complete model for the $3^2 \times 5$ experiment as provided in Table 4.1 of [3] would come out exactly the same by this alternative approach. The procedure of White and Hultquist would require that the number of levels of a factor be prime. But by the procedure presented here, it would be possible to cover mixed factorials of other types where the number of levels of a factor may not be a prime number. Analysis of variance is not attempted here, as such analysis can be carried out following the procedures as given by White and Hultquist [3].

Fisher [5] introduced a combinatorial method to obtain sampling cumulants of $k$-statistics as linear functions of cumulants of an infinite parent population. Kendall [6] systematized Fisher's combinatorial method by providing rules for the same and their proofs. Tukey [11] considered the sample statistic $k_{rs\cdots}$ in order to simplify the presentation of sampling moment formulae of the $k$-statistics when samples are drawn from a finite population. These $k_{rs\cdots}$, termed generalized $k$-statistics by Abdel-Aty [1] and polykays$^2$ by Tukey [12], were in fact considered earlier by Dressel [3] for the seminvariant case $(r, s,\cdots \neq 1)$. Wishart [13] modified the combinatorial method to obtain products of $k$-statistics as linear combinations of polykays, obtained products of polykays by algebraic manipulation, and applied these to the case of a finite population. He provided formulae for products of $k$-statistics through weight 8, and of polykays through weight 6. These have appeared again in David, Kendall and Barton ([2], 196-200, Table 2.3). Schaeffer and Dwyer [8] provided formulae for products of seminvariant polykays through weight 8. Tracy [9] supplied formulae for all products of polykays of weight 7. Dwyer and Tracy [4] modified and extended the combinatorial method to obtain products of two polykays. They presented general formulae resulting from this method for products $k(P)k(Q)$, where $k(P) = k_P = k_{p1\cdots p\pi}$ is a polykay having any weight and weight $(Q) \leqq 4$. Such formulae may be looked upon as rules of multiplication of a polykay by another of weight up to 4. It is the purpose of this paper to extend these results to the case of weight $(Q) = 5$. The formulae are presented together for compactness in a tabular form in Table 1, each column of which reads a formula for some $Q$. Checks indicated in Section 4 are applied more easily in the tabular presentation. Illustrations showing the use of the formulae appear in Tables 2 and 3.